**Differentiable** vs. **continuous functions** are foundational concepts in calculus that explore the behaviors of **functions** across **real numbers.**

A **function** is said to be **continuous** at a point if small **changes** in the input near that point result in small **changes** in the output; formally, ( f ) is **continuous** at ( x = a ) if **$\lim_{x \to a} f(x) = f(a)$**.

For a **function** to be **differentiable** at a point, it must not only be **continuous** there but also have a defined **derivative, signifying** a clear **tangent** at that **point.**

Understanding these concepts helps me, and fellow **mathematicians** and **scientists,** predict and describe the intricate workings of **various physical phenomena.** Let’s explore how these pieces fit together and why they’re more important than you might initially think.

## Main Differences Between Differentiable and Continuous Functions

The main differences between **differentiable** and **continuous** functions hinge on their behavior and requirements at a given point or over an interval. **Differentiable** functions must have a defined **slope** or **tangent** at every point, while **continuous** functions need not exhibit such **smoothness**.

**Continuity**: A function is**continuous at a point**if the left-hand limit, right-hand limit, and the value of the function at that point all agree, denoted as $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$.**Continuous**functions have no**discontinuities**such as**jumps**,**holes**, or**oscillatory behavior**, and the graph can be drawn without lifting the pen.**Differentiability**: A function is**differentiable at a point**if there is a well-defined**tangent line**at that point, implying a unique**slope**. This means the limit of the**slope**of the secant lines, as the two points converge on the point of interest, exists and is finite. The**derivative**, $f'(c)$, represents this**slope**and is given by $\lim_{h \to 0} \frac{f(c+h) – f(c)}{h}$, if this limit exists.**Interrelation**: All**differentiable**functions are**continuous**, but not all**continuous**functions are**differentiable**. For instance, $|, x, |$ is**continuous everywhere**but not**differentiable at**$x = 0$ due to a**corner**.

Unlike **continuity**, **differentiability** is a stricter condition requiring both a lack of **discontinuities** and the existence of a **finite, well-defined, instantaneous rate of change** at every point in its domain.

**Piecewise-defined** functions, **graphs** with **corners** or **cusps**, and functions with a **vertical tangent** (like $y = x^{1/3}$ at $x = 0$) illustrate functions that challenge **differentiability** while maintaining **continuity**.

The **Mean Value Theorem**, linking **differentiability** and **continuity**, states that for a **continuously differentiable function**, there is a point where the **tangent** is parallel to the secant line between two endpoints of any interval.

## Conclusion

In my exploration of the relationship between **differentiability** and **continuity**, I’ve observed a fundamental connection: while every **differentiable** function is **continuous**, not every **continuous** function is **differentiable**.

Consider the comparison of these two **mathematical** properties in the context of their place in **calculus.**

A function $f$ is **continuous** at a point $x=a$ if **$\lim_{x \to a} f(x) = f(a)$.** This means the **function’s** limit at **$a$** equals the **function’s** value at** $a$**.

Meanwhile, **differentiability** goes a step further — a function $f$ is **differentiable** at **$x=a$** if the derivative **$f'(a)$** exists, implying that the function has a defined slope at that point.

An example to **reinforce** this knowledge is the **absolute value function $f(x) = |x|$**, which is **continuous** everywhere but **differentiable** everywhere except at **$x = 0$**. Here, the sharp **“corner”** at the **origin** creates a point where the **slope** isn’t defined, highlighting the distinction between **continuity** and **differentiability**.

I encourage you to think of **continuity** as a necessary condition for **differentiability**—a smooth, uninterrupted path.

Yet, **differentiability** is a **higher-order** condition, requiring the path to not only be smooth but also to have a **tangent** line at every point within the **function’s domain.**

Remembering these crucial differences will serve you well in the study of calculus and your **mathematical endeavors.**